| L(s) = 1 | + (−7.10 + 1.25i)5-s + (−3.36 − 2.82i)7-s + (6.85 + 1.20i)11-s + (19.7 + 7.18i)13-s + (21.7 − 12.5i)17-s + (11.6 − 20.1i)19-s + (17.3 + 20.6i)23-s + (25.4 − 9.24i)25-s + (3.00 + 8.26i)29-s + (−23.8 + 19.9i)31-s + (27.4 + 15.8i)35-s + (−13.9 − 24.1i)37-s + (−10.2 + 28.2i)41-s + (8.18 − 46.4i)43-s + (33.0 − 39.3i)47-s + ⋯ |
| L(s) = 1 | + (−1.42 + 0.250i)5-s + (−0.480 − 0.403i)7-s + (0.623 + 0.109i)11-s + (1.51 + 0.552i)13-s + (1.28 − 0.739i)17-s + (0.611 − 1.05i)19-s + (0.754 + 0.898i)23-s + (1.01 − 0.369i)25-s + (0.103 + 0.285i)29-s + (−0.767 + 0.644i)31-s + (0.783 + 0.452i)35-s + (−0.376 − 0.652i)37-s + (−0.250 + 0.688i)41-s + (0.190 − 1.08i)43-s + (0.703 − 0.838i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31706 - 0.0766422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31706 - 0.0766422i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (7.10 - 1.25i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (3.36 + 2.82i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-6.85 - 1.20i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-19.7 - 7.18i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-21.7 + 12.5i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-11.6 + 20.1i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-17.3 - 20.6i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-3.00 - 8.26i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (23.8 - 19.9i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (13.9 + 24.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (10.2 - 28.2i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-8.18 + 46.4i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-33.0 + 39.3i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 37.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-19.7 + 3.47i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-74.5 - 62.5i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-70.1 - 25.5i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-51.3 + 29.6i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (37.3 - 64.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-90.4 + 32.9i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (42.2 + 115. i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-17.3 - 9.99i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (2.12 - 12.0i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.49051681722490534228073766491, −10.66914260550414055601175777620, −9.394420565799446901892493610743, −8.573319992407687072528212289810, −7.32552339845646209234752203246, −6.89242828314315527778643485411, −5.35026582922840278793326018388, −3.87734147295948934566728358513, −3.34237672607241571838929528118, −0.925982531413226858143391560360,
1.01346569547076141964961106164, 3.35621160724425528043573823049, 3.93811101883180085663254896384, 5.52915711136011996908900544329, 6.51557300957560850183813408609, 7.894842855859080113110818149651, 8.345035033135881835165703135884, 9.460149197047347911026723158702, 10.65513015261148848993272727630, 11.47835878780317713080874269283
